The system I use and teach is the old central-east European style of mediants.

Basically, there is Tonic, its two Dominants on the perfect fifths (SubDominant: SD, and SuperDominant: D)
and for each of them there are Mediants on thirds (SubMediants: SM; and SuperMediants: M). We never use "super", it is just dominant and mediant.
Mediants can be diatonic, or chromatic.
Example:
A submediant to subdominant (in C major) is chord d-minor.
A low (=chromatically lowered) submediant to the SubDominant is the Phrygian chord (F) that is Dflat-major, when inverted it is Napolitan (N, usually thus N6).
P is the polar chord, the only one with no function (or semi-T+semi-S function): F# in C-major, thus, if p it is minor if P it is major: